Prepare Bode diagram curves for the second-order system described by

$\frac{d^2y}{dt^{2} } +2\xi \omega _{n}\frac{dy}{dt} +\omega ^{2}_{n}y= \omega ^{2}_{n}x$ . (12.47)

Question

Prepare Bode diagram curves for the second-order system described by

[latex] \frac{d^2y}{dt^{2} } +2\xi \omega _{n}\frac{dy}{dt} +\omega ^{2}_{n}y= \omega ^{2}_{n}x [/latex]. (12.47)

Accepted Answer

Step 1. The system transfer function is

[latex] T(s)=\frac{Y(s)}{X(s)} =\frac{\omega ^{2}_{n}}{x^{2}+2\xi \omega _{n}s+\omega ^{2}_{n} } [/latex]. (12.48)

Step 2. The sinusoidal transfer function is

[latex] T(j\omega ) =\frac{\omega ^{2}_{n}}{(j\omega )^{2}+2\xi \omega _{n}j\omega +\omega ^{2}_{n} }=\frac{1}{1-({\omega }/{\omega _{n} })^{2}+j2\xi ({\omega }/{\omega _{n} }) } [/latex]. (12.49)

Step 3. Develop expressions for T(ω) and [latex] \phi_{T} (ω)[/latex]. First, the magnitude is

[latex] T(j\omega ) =\frac{1}{\sqrt{\left[1-({\omega }/{\omega _{n} })^{2} ight] ^{2}+4\xi ^{2}({\omega }/{\omega _{n} })^{2}}} [/latex]. (12.50)

Then the phase angle is

[latex] \phi _{T} (\omega ) =- an ^{-1} \frac{2\xi({\omega }/{\omega _{n} }) }{1-({\omega }/{\omega _{n} })^{2} } [/latex]. (12.51)

Step 4. Prepare the Bode diagram curves, as shown in Fig. 12.6.

Question 12.3: Prepare Bode diagram curves for the second-order system desc...

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